Simplify the following expression and state the condition under which the simplification is valid. $z = \dfrac{-9k^3 - 153k^2 - 630k}{-8k^2 - 104k - 336}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ z = \dfrac {-9k(k^2 + 17k + 70)} {-8(k^2 + 13k + 42)} $ $ z = \dfrac{9k}{8} \cdot \dfrac{k^2 + 17k + 70}{k^2 + 13k + 42} $ Next factor the numerator and denominator. $ z = \dfrac{9k}{8} \cdot \dfrac{(k + 7)(k + 10)}{(k + 7)(k + 6)}$ Assuming $k \neq -7$ , we can cancel the $k + 7$ $ z = \dfrac{9k}{8} \cdot \dfrac{k + 10}{k + 6}$ Therefore: $ z = \dfrac{ 9k(k + 10)}{ 8(k + 6)}$, $k \neq -7$